Solitons trapping for the nonlinear Klein–Gordon equation with an external excitation

The asymptotic perturbation (AP) method is applied to the study of the nonlinear Klein–Gordon equation and an external periodic excitation is supposed to be in primary resonance with the frequency of a generic mode. The AP method uses two different procedures for the solutions: introducing an asymptotic temporal rescaling and balancing of the harmonic terms with a simple iteration. Amplitude and phase modulation equations and external force–response and frequency–response curves are obtained and the presence of multistabilities and solitons trapping is demonstrated. The validity of the method is highlighted by comparing the approximate solutions with results of the numerical integration.